Final answer:
The sampling distribution of the sample mean, x, is approximately normally distributed. To find the probabilities in the given problem, we need to standardize the values using the z-score formula. The probability of x>79 is approximately 0.0668, the probability of x≤71.3 is approximately 0.0094, and the probability of x=73 is approximately 0.68.
Step-by-step explanation:
(a) Describe the sampling distribution of x:
The sampling distribution of x, the sample mean, is approximately normally distributed, particularly if the sample size is large (>30) and the population distribution is not extremely skewed or has outliers.
(b) What is P(x>79)?:
To find P(x>79), we need to standardize the value using the formula z = (x - mean) / standard deviation. Then, we can look up the z-score in the standard normal distribution table or use a calculator to find the probability. In this case, we would calculate z = (79 - 76) / (12 / sqrt(36)) = 3 / 2 = 1.5. Looking up the z-score, we find that the probability is approximately 0.0668.
(c) What is P(x≤71.3)?:
To find P(x≤71.3), we again need to standardize the value. Using the formula z = (x - mean) / standard deviation, we get z = (71.3 - 76) / (12 / sqrt(36)) = -4.7 / 2 = -2.35. Looking up the z-score, we find that the probability is approximately 0.0094.
(d) What is P(73)?:
Since P(73) is a single value, it does not need to be standardized. We can use the empirial rule, also known as the 68-95-99.7 rule, to estimate the probability. According to this rule, approximately 68% of the data falls within one standard deviation of the mean. So, the probability of getting a value of 73 is approximately 0.68.